Maximum Likelihood Reconstruction for Emission Tomography

Shepp, L. A.; Vardi, Y.

doi:10.1109/tmi.1982.4307558

Cited by 4,429Medical Imaging Techniques and ApplicationsNumerical methods in inverse problemsImage and Signal Denoising Methods

Maximum Likelihood Reconstruction for Emission Tomography

Shepp, L. A., Vardi, Y.·1982·IEEE Transactions on Medical Imaging

DOI 10.1109/tmi.1982.4307558

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Abstract

Previous models for emission tomography (ET) do not distinguish the physics of ET from that of transmission tomography. We give a more accurate general mathematical model for ET where an unknown emission density lambda = lambda(x, y, z) generates, and is to be reconstructed from, the number of counts n(*)(d) in each of D detector units d. Within the model, we give an algorithm for determining an estimate lambdainsertion mark of lambda which maximizes the probability p(n(*)|lambda) of observing the actual detector count data n(*) over all possible densities lambda. Let independent Poisson variables n(b) with unknown means lambda(b), b = 1, ..., B represent the number of unobserved emissions in each of B boxes (pixels) partitioning an object containing an emitter. Suppose each emission in box b is detected in detector unit d with probability p(b, d), d = 1, ..., D with p(b,d) a one-step transition matrix, assumed known. We observe the total number n(*) = n(*)(d) of emissions in each detector unit d and want to estimate the unknown lambda = lambda(b), b = 1, ..., B. For each lambda, the observed data n(*) has probability or likelihood p(n(*)|lambda). The EM algorithm of mathematical statistics starts with an initial estimate lambda(0) and gives the following simple iterative procedure for obtaining a new estimate lambdainsertion mark(new), from an old estimate lambdainsertion mark(old), to obtain lambdainsertion mark(k), k = 1, 2, ..., lambdainsertion mark(new)(b)= lambdainsertion mark(old)(b)Sum of (n(*)p(b,d) from d=1 to D/Sum of lambdainsertion mark()old(b('))p(b('),d) from b(')=1 to B), b=1,...B.

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Estimates of I maximizing Eq. 1 obtained with an expectation maximization algorithm are typically unstable.
“Estimates of I maximizing Eq. 1 can be obtained with an expectation maximization algorithm (6, 33–35), but the outcome is typically unstable (36).”
From Socioeconomic status determines COVID-19 incidence and related mortality in Santiago, Chile (2021) · cites this paper as [33]

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Authors: Shepp, L. A., & Vardi, Y.
Affiliations: AT&T (United States) (US)
Venue: IEEE Transactions on Medical Imaging
Publisher: Institute of Electrical and Electronics Engineers (IEEE)
Published: 1982-10
Volume: 1
Issue: 2
Pages: 113-122
ISSN: 0278-0062 (ISSN-L)
1558-254X
DOI: 10.1109/tmi.1982.4307558
OpenAlex: W2069629287
Cited by: 4,429
References: 9
Citations by year: 2026: 74 · 2025: 101 · 2024: 122 · 2023: 101 · 2022: 117 · 2021: 140 · 2020: 154 · 2019: 153 · 2018: 137 · 2017: 142 · 2016: 152 · 2015: 164 · 2014: 167 · 2013: 182 · 2012: 176
FWCI: 10.00
Type: journal article
Language: en
Open access: closed
Licence: ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html (vor)
Locations: IEEE Transactions on Medical Imaging (publishedVersion)
PubMed (publishedVersion)
Topics: Health Sciences › Medicine › Radiology, Nuclear Medicine and Imaging › Medical Imaging Techniques and Applications
Physical Sciences › Mathematics › Mathematical Physics › Numerical methods in inverse problems
Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Image and Signal Denoising Methods
Keywords: emission tomography, maximum likelihood reconstruction, expectation-maximization algorithm, Poisson likelihood, tomographic reconstruction

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Maximum Likelihood Reconstruction for Emission Tomography

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